Table of Integrals

College/University

Definition

Helps with integration in calculus. The table of integrals lists out common antiderivatives so that you can carry out calculations that involve integrations. It also lists out the integrals in different categories to make it easier for you to find the ones you'll need.

Worked examples

\(\int \sin(x) \, dx = -\cos(x) + C\)

Look up the integral of sine in the table rather than deriving it each time.

\(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\)

Tables list formulas for exponentials with coefficients so you can substitute your value of a directly.

\(\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\)

Rational-function integrals are grouped by denominator pattern in the table.

Common mistakes

Using \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) when \(n = -1\) → \(\int \frac{1}{x} \, dx = \ln|x| + C\) The power rule fails at n = -1; the table lists this case separately as the natural logarithm.
Forgetting the constant \(C\) when copying from the table → Always include \(+ C\) for indefinite integrals Every antiderivative family differs by a constant; omitting C loses infinitely many solutions.
Applying a table formula without checking that your integral matches exactly → Use substitution or algebra first to transform your integral into table form Tables assume standard forms; you must manipulate your integrand to match before using a formula.

Where you'll use it next

You'll rely on integral tables throughout Calculus II for techniques of integration, in differential equations to solve separable and linear equations, and in physics and engineering for computing work, center of mass, and Fourier transforms.

Found in 1 StudyPug lesson

Antiderivatives

Calculus 2

In this section, we will examine closely the difference between a derivative and an anti-derivative. Always remember that the anti-derivative has a constant of integration. Once we understand the concept of anti-derivatives, we will look at the anti-derivative of polynomials and anti-derivative of rational functions. We will then take a look at harder functions such as irrational functions and trigonometric functions. Once we have a general understanding of the concept, we will actually find the constant of integration using initial conditions. Lastly, we will apply anti-derivatives to real life applications such as position, velocity, and acceleration.

UniversityUniversity